我們提出了一個電偶極子在金納米柱雙曲線超材料（HMM）薄膜上的模態的理論分析。在長波長區域，我們將雙曲線超材料簡化為有效的各向異性介質，並且表明最優化的Purcell 因子源於TM 模式反射係數的虛部，其不僅由材料的雙曲散色射特性確定，而且和由於法布里 - 珀羅共振決定的厚度的選擇性厚度相關。我們將雙曲線超材料薄膜內的平面波傳播描述為“臨界耦合模式”和“共振模式”，並從空間頻域分布的角度解釋電偶極子場增強的現象。我們細致討論了HMM薄膜厚度，金屬填充率，材料的吸收系數和電偶極鉅變小導致湮沒等因素。通過嚴格耦合波分析理論優化出幾何結構，我們設計了一個矽同心靶光柵，並展示了12 倍以上的遠場輻射增強。這種數值方法不僅適用於納米柱HMM材料，而且適用於多層HMM材料和其他各向異性介質，是作為設計高速非相干光源器件的有效數值工具。

We present an analytical description of the modes property for a dipole emitter when put above a gold nanorods hyperbolic metamaterials slab. In the long wavelength region, we simplify the hyperbolic metamaterials as an effective anisotropic medium and show that the optimized purcell factor originates from the imaginary part of the TM mode reflection coefficients, which is not only determined by bulk hyperbolic dispersion but also the selective slab thickness due to the Fabry-Perot resonance. We characterize the plane wave propagation inside the HMM slab as ‘critical coupling mode’ and ‘resonating mode’ and explain the dipole field enhancement in a view of angular expansion. Factors of HMM slab thickness, the metal fill ratio, metal loss and quenching effect are taken into discussion. Using the rigorous coupled wave analysis with the optimized geometry, we deliberately design a silicon bullseye grating and demonstrate more than 12 folds farfield radiation enhancement. This numerical method is not only suitable for nanorods HMM material but also for multilayer HMM material and other anisotropic medium, serving as an efficient tool for the design of high speed incoherent optical source devices.

Chapter 1 Introduction
1.1 Hyperbolic Metamaterials(HMM) ………………………………………………1
1.1.1 One Dimensional HMM ………………………………………………………1
1.1.2 Two Dimensional HMM ………………………………………………………3
1.2 Density of States Engineering …………………………………………………5
1.2.1 Purcell Factor and Farfield Radiation Enhancement……………………………5
1.2.2 Literature Review ……………………………………………………………6
1.2.3 Our Work ……………………………………………………………………13
Chapter 2 Numerical Method
2.1 Effective Medium Approximation ………………………………………………14
2.2 Angular Spectrum Representation of Dipole Field ……………………………15
2.2.1 Dipole Field: TE Polarization …………………………………………………18
2.2.2 Dipole Field: TM Polarization ………………………………………………21
2.3 Rigorous Coupled Wave Analysis ………………………………………………25
2.3.1 RCWA: TE Polarization ……………………………………………………26
2.3.2 RCWA: TM Polarization ……………………………………………………29
2.4 Double Exponential Rule Integral for Sommerfeld Tails ………………………33
Chapter 3 Results and Discussion
3.1 Wave Propagation: Dielectric to Infinite Type I HMM …………………………36
3.2 Wave Propagation: Dielectric - Type I HMM Slab - Dielectric Schematic ……38
3.2.1 Density of States with Fixed Fill Ratio ………………………………………40
3.2.2 Optimized Thickness: Critical Coupling and Resonating ……………………42
3.2.3 The Effect of Loss ……………………………………………………………50
3.2.4 Varying the Fill Ratio …………………………………………………………53
3.2.5 Quenching ……………………………………………………………………57
3.3 Outcoupling the High-k Modes into Farfield …………………………………59
3.3.1 Grating Coupling: Plane Wave Incidence ……………………………………61
3.3.2 Bullseye Grating optimization………………………………………………62
Chapter 4 Conclusion and Outlook
4.1 Advantages and Disadvantages …………………………………………………63
4.2 Future Improvements …………………………………………………………64
References …………………………………………………………………………65
Appendix …………………………………………………………………………68

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